Average Error: 17.0 → 3.8
Time: 41.9s
Precision: binary64
\[[phi1, phi2]=\mathsf{sort}([phi1, phi2])\]
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[e^{\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) - 1\right)} \cdot R \]
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R
↓
e^{\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) - 1\right)} \cdot R

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))
↓
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (exp
   (log1p
    (-
     (acos
      (fma
       (cos phi2)
       (*
        (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2)))
        (cos phi1))
       (* (sin phi1) (sin phi2))))
     1.0)))
  R))
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda1 - lambda2))) * R;
}
↓
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return exp(log1p(acos(fma(cos(phi2), (fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))) * cos(phi1)), (sin(phi1) * sin(phi2)))) - 1.0)) * R;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 17.0

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

  2. Applied cos-diff_binary643.7

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]

  3. Applied add-exp-log_binary643.7

    \[\leadsto \color{blue}{e^{\log \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}} \cdot R \]

  4. Simplified3.7

    \[\leadsto e^{\color{blue}{\log \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)}} \cdot R \]

  5. Applied log1p-expm1-u_binary643.8

    \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\right)}} \cdot R \]

  6. Taylor expanded in phi1 around -inf 3.8

    \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) - 1}\right)} \cdot R \]

  7. Final simplification3.8

    \[\leadsto e^{\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) - 1\right)} \cdot R \]

Reproduce

herbie shell --seed 2021291 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))